In the framework of automated manufacturing systems (AMS), Petri nets are widely used to model, analyze, and control them. Resolving deadlocks is of paramount significance because their emergence may zero a system's throughput. Supervisory control technique is the most widely adopted method to resolve them. A control policy can be converted into satisfying a set of inequalities, each of which corresponds to a siphon in a Petri net structure. The number of siphons can be exponential in the worst case, so does the number of inequalities. Taking into account the independent and dependent inequalities, this paper proposes a method to remove all the dependent inequalities while preserving only the independent ones. This method can significantly reduce the size of a supervisory controller. Examples are presented to illustrate the effectiveness and efficiency of this method.Note To Practitioners-Owing to its importance in practice, deadlock arouses interest. Various resolution methods have been proposed by researchers and practitioners. The recent advance is to associate various sensors and actuators to certain predetermined working nodes for the sake of management and coordination of limited resources among concurrent operations. This uses a supervisory controller to a plant model. Nevertheless, existing approaches suffer from structural complexity due to their failure to reduce the sophisticated supervisor structures, which impedes their implementation in practice. Without a simplified control network, the communication cost among heterogeneous nodes increases while the responsive time decreases. Through the analysis on the dependency relationship among different inequalities, this paper presents a reduction technique to significantly simplify the supervisor structures. Without any extra investment, our approach betters a controlled system's performance.

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Calcium binding proteins are well known to be expressed by different groups of hippocampal interneurons; however, whether voltage-dependent calcium channels (Ca(v)) are also localized in these neurons, changed during and after status epilepticus (SE), and involved in epileptic activity have not been reported. In the present study, we showed the colocalization of three subtypes of voltage-gated calcium channels (Ca(v)1.2, Ca(v)1.3, or Ca(v)2.1) with different calcium binding proteins such as calbindin (CB), calretinin (CR), and parvalbumin (PV). At early stages during and after pilocarpine-induced status epilepticus (PISE), significant changes of expression of Ca(v)1.2, Ca(v)1.3 (L-type), and Ca(v)2.1 (P/Q-type) were found in different groups of hippocampal neurons. Induced expression of Ca(v)1.3 or Ca(v)2.1 in reactive astrocytes was shown at 1 week and 2 months after PISE. At the latter time point, higher percentages of colocalization of PV and Ca(v)1.2, CB, or PV and Ca(v)1.3 or Ca(v)2.1, lower percentages of CR and Ca(v)1.3 or Ca(v)2.1 immunoposivie neurons were observed in gliotic CA1 area. We therefore conclude that voltage-gated calcium channels are expressed by different groups of hippocampal interneurons in the mouse. At acute stages during and after PISE, up- or down-regulation of Ca(v)1.2, Ca(v)1.3, or Ca(v)2.1 in functionally different groups of interneurons in CA1 area may be related to the changes of their plasticity. Up-regulation of Ca(v)1.2, Ca(v)1.3, or Ca(v)2.1 in granule cells may be directly related to the occurrence of SE. The induced expression of Ca(v)1.3 or Ca(v)2.1 in reactive astrocytes at 1 week and 2 months after PISE suggests that Ca(v)1.3 or Ca(v)2.1-related calcium signaling in reactive astrocytes may be involved in initiation, maintenance or spread of seizure activity. In gliotic CA1 area at chronic stage (i.e., 2 months after PISE), the occurrence of higher percentages of colocalization of PV and Ca(v)1.2, CB, or PV and Ca(v)1.3 or Ca(v)2.1, lower percentages of CR and Ca(v)1.3 or Ca(v)2.1 immunopositive neurons may suggest that such colocalizations may be linked to the survival or loss of particular group of hippocampal neurons.

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